Subgroup decomposition in Out(F_n), Part I: Geometric Models

Abstract

This is the first in a series of four papers, announced in [HM13a], that together develop a decomposition theory for subgroups of Out(F n ).In this paper we develop further the theory of geometric EG strata of relative train track maps originally introduced in [BFH00] Section 5, with our focus trained on certain 2-dimensional models of such strata called "geometric models" and on the interesting properties of these models. A secondary purpose of this paper is to serve as a central reference for the whole series regarding basic facts of the general theory for elements of Out(F n ).

The free splitting complex of a free group, II: Loxodromic outer automorphisms

“…The meet operation on pairs extends to a well-defined operation on any set of free factor systems. To each marked graph G and subgraph K there corresponds a free factor system denoted F(K) or [K], by taking the conjugacy classes of the subgroups of π 1 (G) ≈ F n corresponding to the noncontractible components of K. See Section 2.6 of [BFH00], and Sections 1.1.2 and 3.1 of [HM13c].…”

“…A set of lines fills F n if its free factor support is {[F n ]}. For each subgraph K of a marked graph G the free factor system F(K) is the free factor support of the set of conjugacy classes represented by circuits in K. See Section 2.6 of [BFH00] or Section 2.5 of [FH11] or Sections 1.1.2 and 1.2.2 of [HM13c].…”

“…After passing to a positive power, subdividing certain edges, and straightening f the following properties hold: any NEG stratum H i consists of a single oriented edge E that satisfies f (E) = E • u for some (possibly trivial) path in G i−1 ; and any EG stratum H r has the property that for all sufficiently large k the f k # -image of each edge in H r crosses every edge in H r and the number of such crossings grows exponentially in k. Furthermore, any EG stratum has ≥ 2 edges. See Section 5 of [BH92], Subsection 1.5.1 of [HM13c] or Subsections 2.6 and 2.7 of [FH11].…”

“…We say that f : G → G is rotationless if each principal vertex v ∈ G is fixed and each periodic direction at v has period one. See Definition 3.18 of [FH11] and Section 1.5.1 of [HM13c] for further discussion.…”

“…In this section we set notation and provide references to [BFH00], [FH11] and [HM13c] for readers that want further details.…”